Optimal. Leaf size=121 \[ \frac {81}{520} (1-2 x)^{13/2}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {2 (1-2 x)^{5/2}}{15625}+\frac {22 (1-2 x)^{3/2}}{46875}+\frac {242 \sqrt {1-2 x}}{78125}-\frac {242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \]
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Rubi [A] time = 0.04, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {88, 50, 63, 206} \[ \frac {81}{520} (1-2 x)^{13/2}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {2 (1-2 x)^{5/2}}{15625}+\frac {22 (1-2 x)^{3/2}}{46875}+\frac {242 \sqrt {1-2 x}}{78125}-\frac {242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{3+5 x} \, dx &=\int \left (\frac {136419 (1-2 x)^{5/2}}{5000}-\frac {34371 (1-2 x)^{7/2}}{1000}+\frac {2889}{200} (1-2 x)^{9/2}-\frac {81}{40} (1-2 x)^{11/2}+\frac {(1-2 x)^{5/2}}{625 (3+5 x)}\right ) \, dx\\ &=-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {81}{520} (1-2 x)^{13/2}+\frac {1}{625} \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx\\ &=\frac {2 (1-2 x)^{5/2}}{15625}-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {81}{520} (1-2 x)^{13/2}+\frac {11 \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx}{3125}\\ &=\frac {22 (1-2 x)^{3/2}}{46875}+\frac {2 (1-2 x)^{5/2}}{15625}-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {81}{520} (1-2 x)^{13/2}+\frac {121 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{15625}\\ &=\frac {242 \sqrt {1-2 x}}{78125}+\frac {22 (1-2 x)^{3/2}}{46875}+\frac {2 (1-2 x)^{5/2}}{15625}-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {81}{520} (1-2 x)^{13/2}+\frac {1331 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{78125}\\ &=\frac {242 \sqrt {1-2 x}}{78125}+\frac {22 (1-2 x)^{3/2}}{46875}+\frac {2 (1-2 x)^{5/2}}{15625}-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {81}{520} (1-2 x)^{13/2}-\frac {1331 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{78125}\\ &=\frac {242 \sqrt {1-2 x}}{78125}+\frac {22 (1-2 x)^{3/2}}{46875}+\frac {2 (1-2 x)^{5/2}}{15625}-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {81}{520} (1-2 x)^{13/2}-\frac {242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 71, normalized size = 0.59 \[ \frac {5 \sqrt {1-2 x} \left (2338875000 x^6+2842087500 x^5-1540428750 x^4-2556079875 x^3+399578370 x^2+960784285 x-289133384\right )-726726 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1173046875} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 76, normalized size = 0.63 \[ \frac {121}{390625} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac {1}{234609375} \, {\left (2338875000 \, x^{6} + 2842087500 \, x^{5} - 1540428750 \, x^{4} - 2556079875 \, x^{3} + 399578370 \, x^{2} + 960784285 \, x - 289133384\right )} \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 138, normalized size = 1.14 \[ \frac {81}{520} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {2889}{2200} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {3819}{1000} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {136419}{35000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {2}{15625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {22}{46875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{390625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {242}{78125} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 83, normalized size = 0.69 \[ -\frac {242 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{390625}+\frac {22 \left (-2 x +1\right )^{\frac {3}{2}}}{46875}+\frac {2 \left (-2 x +1\right )^{\frac {5}{2}}}{15625}-\frac {136419 \left (-2 x +1\right )^{\frac {7}{2}}}{35000}+\frac {3819 \left (-2 x +1\right )^{\frac {9}{2}}}{1000}-\frac {2889 \left (-2 x +1\right )^{\frac {11}{2}}}{2200}+\frac {81 \left (-2 x +1\right )^{\frac {13}{2}}}{520}+\frac {242 \sqrt {-2 x +1}}{78125} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 100, normalized size = 0.83 \[ \frac {81}{520} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {2889}{2200} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {3819}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {136419}{35000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {2}{15625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {22}{46875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{390625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {242}{78125} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 84, normalized size = 0.69 \[ \frac {242\,\sqrt {1-2\,x}}{78125}+\frac {22\,{\left (1-2\,x\right )}^{3/2}}{46875}+\frac {2\,{\left (1-2\,x\right )}^{5/2}}{15625}-\frac {136419\,{\left (1-2\,x\right )}^{7/2}}{35000}+\frac {3819\,{\left (1-2\,x\right )}^{9/2}}{1000}-\frac {2889\,{\left (1-2\,x\right )}^{11/2}}{2200}+\frac {81\,{\left (1-2\,x\right )}^{13/2}}{520}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,242{}\mathrm {i}}{390625} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 103.66, size = 150, normalized size = 1.24 \[ \frac {81 \left (1 - 2 x\right )^{\frac {13}{2}}}{520} - \frac {2889 \left (1 - 2 x\right )^{\frac {11}{2}}}{2200} + \frac {3819 \left (1 - 2 x\right )^{\frac {9}{2}}}{1000} - \frac {136419 \left (1 - 2 x\right )^{\frac {7}{2}}}{35000} + \frac {2 \left (1 - 2 x\right )^{\frac {5}{2}}}{15625} + \frac {22 \left (1 - 2 x\right )^{\frac {3}{2}}}{46875} + \frac {242 \sqrt {1 - 2 x}}{78125} + \frac {2662 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 < - \frac {11}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 > - \frac {11}{5} \end {cases}\right )}{78125} \]
Verification of antiderivative is not currently implemented for this CAS.
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